An Introduction to Plane Curves

A plane curve is any curve which can be drawn on the plane. Some curves are fairly simple, like a circle, and will have fairly simple algebraic equations. Some are very complex, like your signature, and may be very difficult to describe with an equation. Plane curves were studied intensively from the seventeenth through the nineteenth centuries. Current mathematical research and coursework is often much more generalized and abstract than the items you will find on this web site.

The equation for any given plane curve will depend on the coordinate system used. Three common coordinate systems and their generic plane curve equations include:

rectangular, $f(x,y)=0$
polar, $f(r,\theta)=0$
parametric, $x=f(t), y=g(t)$

Plane curves will also arise from the level curves, or contour curves, of functions of two independent variables. Functions of two variables will produce three-dimensional graphs of ordered triples, $(x,y,z)$, where $z=f(x,y)$.

The simplest plane curves arise from algebraic equations. An equation is algebraic when the only operations involved are addition, subtraction, multiplication, division, extraction of roots, and raising to integral or fractional powers. Algebraic functions of one variable can be written as polynomial equations in two variables. They are often classified as linear (lines), quadratic (conic sections), or higher plane curves (using polynomials of degree 3 or more).

Equations which are not algebraic are called transcendental equations. The simplest transcendental equations will involve exponential, logarithmic, trigonometric, or hyperbolic functions (all of which are related in the complex number system). Other examples of transcendental functions would include the Gamma function, Bessel functions, and the Error function of statistics.

Plane curves can exhibit many different interesting features. Some of these features include:

multiple branches
asymptotes, either linear or nonlinear
isolated points
nodes (where a curve crosses itself)
cusps (where a curve is tangent to itself)

Some plane curves are degenerate curves. Degeneracy occurs when a curve is both a simpler type and a limiting case of a family of curves. Algebraically, this often occurs when some coefficient or parameter vanishes. Two examples include:

a single point as a degenerate circle (when the radius of a circle shrinks to zero)
two intersecting lines as a degenerate hyperbola (when the constant term of the algebraic equation reaches the value where the equation becomes factorable as the difference of two squares)

An advanced study of plane curves would introduce the projective plane and homogeneous coordinates. The projective plane is essentially the Euclidean plane with a "line at infinity" added. Lines which are parallel in the Euclidean plane have a point of intersection on the line at infinity. Homogeneous coordinates allow us to describe all points in the projective plane with a family of ordered triples of real numbers (infinity is never used as a coordinate). By using the projective plane, many different-looking curves can be seen to be related (for example, the circle and the hyperbola). But on this web site, we will generally not make these connections. To learn more, you should take a course in algebraic geometry.